INTRODUCTION

High speed communications channels are often impaired by channel inter-symbol interference, co-channel interference and additive noise. Adaptive equalizers are required in these communication systems to obtain reliable data recovery [1].

The discrete-time model of the digital communications systems considered in this paper is depicted in figure 1. In this model H₀(z) is the channel transfer function and there are a total of ${\eta}$ interfering co-channels with transfer functions ${H_i(z),{1\leq i \leq\eta}}$. The linear dispersive channel and co-channels are modeled by finite impulse response filters and, therefore, their transfer functions are given by

$$H_i(z)=\sum_{j=0}^{n_s} h_{ij} z^{-j}.\hspace{1 in} 0 \leq i \leq \eta$$ (1)

The transmitted data d₀(k) and the interfering data $d_i(k), 1 \leq i \leq \eta$ are assumed to be equiprobable and independent sequences. The data $d_i(k), 0 \leq i \leq \eta$, would have zero mean, i.e., E[d_i(k)] = 0 and $E[d_i(k_1)d_j(k_2)]=\delta(i-j) \delta(k_1 - k_2)$, where E[.] denotes the expectation operator. The additive white Gaussian noise e(k) has zero mean and variance $\sigma_e^2$ and is uncorrelated with the data $d_i(k), 0 \leq i \leq \eta$.

As defined in [2] the channel observation y(k) = s(k) + u(k) + e(k) contains three terms called the desired signal, the interfering signal and the noise, respectively: $s(k)=\sum_{j=0}^{n_0} h_{0j} d_0(k-j)$ and $u(k)=\sum_{i=1}^{\eta}\sum_{j=0}^{n_i}h_{ij}d_i (k-j)$. Let $E[s^2(k)] = \sigma_s^2$ and $E[u^2(k)] = \sigma_u^2$. The signal to noise ratio is then defined as $SNR = {\sigma_s^2}/{\sigma_e^2}$ and the signal to interference ratio is given by $SIR = {\sigma_s^2}/{\sigma_u^2}$ and finally the signal to interference and noise ratio is given by $SINR = {\sigma_s^2}/({\sigma_e^2}+{\sigma_u^2})$ [3].

The task of the equalizer is to estimate the transmitted data d₀(k) based on the channel observation y(k). There are variety of approaches proposed [10], basically dividing equalization into spread-spectrum (wideband) and nonspread spectrum (narrowband) techniques. The popular techniques with nonspread spectrum are the constant modulus algorithm (CMA), neural networks (NN) and higher order statistics (HOS) based algorithms. The NN techniques are sub-classified into those based on radial basis functions, backpropagation and polynomial perceptrons. The multilayer perceptrons however, has problems of slow convergence and unpredictable solutions during the training while the polynomial equalizer suffers from the drawback of having exponentially increasing filter dimensions. In this paper we are looking at the application of radial basis function (RBF) for the purpose of equalization.

Hosted by www.Geocities.ws